How Much Sooner Does a Box Reach the Bottom of an Incline Compared to a Disk?

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The discussion focuses on the comparative analysis of a box and a disk of equal mass rolling down frictionless and frictional inclines, respectively. The box reaches the bottom of the incline faster than the disk due to its different moment of inertia and the absence of friction affecting its motion. The analysis employs conservation of energy principles and considers the variables mass (m), height (h), angle (θ), radius (R), and gravitational acceleration (g). The problem highlights the importance of understanding the effects of friction and rotational inertia on the dynamics of rolling objects.

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A disk and a box of equal mass are released from the top of two inclines both of which are a height h above the ground and make an angle θ to the horizontal. Let the radius of the disk be R. How much sooner does the box reach the bottom of the incline than the disk?
Express your answer in terms of some or all of the variables m, h, theta, and R, as well as the acceleration due to gravity g.
 
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Probably best to tackle this one with conservation of energy and with moments of inertia :)
 
The problem statement doesn't mention a cofficient of friction. Is the inclined plane frictionless for the box and not frictionless for the disk?
 

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